In the field of control of mechanical systems, whether they are servomechanisms or robotized systems, the dry friction in said systems very often constitutes a source of difficulty for the engineer in charge of conceiving feedback control laws.
Indeed, this friction has an intrinsically non-linear behaviour (hard non-linearity), prejudicial to the quality of follow-up of a position, speed, acceleration, effort setpoint instruction, both for translational and rotational motions. This friction may also generate limit cycles (oscillatory phenomena of the position of a mechanical system controlled by a constant setpoint instruction) or produce a “stick and slip” phenomenon, commonly called “grabbing”, causing the mechanical system to move jerkily.
The explication of the origin of such friction pertains to the tribology field. It results from the accumulation of a very high number of phenomena occurring at the microscopic or even atomic scale. Unfortunately, the microscopic approach is not very useful for the automation expert, due to the complexity of the physical phenomena involved that make its exploitation inconceivable for the building of control laws.
In a feedback control loop, the dry friction is hence commonly modelled by a disturbing force (or torque), whose sign is in first approximation opposed to the speed thereof: it is talked about Coulomb friction. To that is added a so-called Stribeck effect: after immobilization, the actuator must overcome a force (torque) called “Stiction” force/torque, whose module is higher than that required to maintain the motion, one the latter is effective.
In the field of automation, the dry friction compensation technics exist since the beginning of the 1940's. One of the first methods intended to attenuate the grabbing phenomenon consisted in additively introducing into the actuator control signal a sinusoidal signal of relatively high frequency with respect to the bandwidth of the feedback control. This technic is called “dither”.
Since the 60's have progressively appeared macroscopic models intended to describe this friction from the phenomenological point of view, with the mathematical tool. These more or less complicated models, describing more or less finely the behaviour of mechanical systems, are called Tustin, Karnopp, Dahl, Slimane-Sorine, LuGre models . . . . The elaboration of such macro-models is still today a subject of research.
In parallel with the development of the friction models, compensation control laws based on said models have progressively appeared, consisting in particular in estimating and compensating the friction torque in real time as a function of the measured speed of the mechanical system: an application of this approach has been made in particular based on the LuGre model by H. Olson, K. J. Aström et al. in the article: “Friction models and friction compensation”, European journal of control, 1998.
To be effective, such friction compensation control laws must be based on accurately parameterized models. Now, the characteristics of the friction of the mechanical system are liable to evolve over time, as a function for example of the temperature, of the lubrication of the parts in contact, of the wear, etc. . . . .
The above-described models have in common to depend only on the speed of the mobile mechanical system. Some of them, too basic, are not simulatable as such (as for example the Coulomb model), and cause high-frequency switches leading to the freezing of the simulation tool. Other ones describe the friction phenomenon far finely, as for example the LuGre model, but the number of parameters constituting it is relatively high (6 parameters for the LuGre model) and the identification of these parameters is a task that may be long to implement, especially in an industrial context in which, in particular, the respect of the time limits is an important criterion.
In 2009, a new dry friction model has been published by Philippe de Larminat in the book “Automatique appliquée, 2nde édition”, Hermès edition. This model, contrary to the preceding ones, has been elaborated starting from the observation that the friction effort is function not only of the speed but also of the force (or torque) delivered by the actuator. This model is, as emphasized by the author, one of the simplest models that has never been developed. It describes the Coulombian effect as well as the Stribeck effect. Moreover, this model has the advantage to have a restricted number of parameters: two parameters for the Coulombian effect and a third one for the Stribeck effect.
Associated with this model, a compensation control law has been proposed in the same book, which also has a simplicity that is very interesting for an implementation in real time in an industrial framework. This control law takes only into account the mass or the inertia of the controlled mobile mechanical system, a minor time constant and a variable representative of the Coulomb module. Nevertheless, to be efficient, this compensation control law presupposes a precise knowledge of the Coulomb module of the friction. Failing that, the control law “overcompensate” or “undercompensate” for the friction and the beneficial effects on the feedback controls of the mechanical system are very reduced. Hence, this dry friction compensation control law, efficient as it is, really risks to have an ephemeral efficiency, because it will become inoperative when the physical parameters of the friction will have evolved over time.
Hence, it is herein proposed to add an adaptive structure to the compensation control law presented in the above-mentioned book of Ph. de Larminat and that is what the present invention proposes.
It is known in this field an article by M. Itthise Nilkhamhang, “Adaptive Friction Compensation using the GMS Model with Polynomial Stribeck function”, which proposes to linearize a GMS model.